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Caracterization of Ergodic Actions and Quantum Analogue of Noether’S Isomorphisms Theorems Souleiman Omar Hoch

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Caracterization of Ergodic Actions and Quantum Analogue of Noether’S Isomorphisms Theorems Souleiman Omar Hoch Study of compact quantum groups with probabilistic methods : caracterization of ergodic actions and quantum analogue of Noether’s isomorphisms theorems Souleiman Omar Hoch To cite this version: Souleiman Omar Hoch. Study of compact quantum groups with probabilistic methods : caracterization of ergodic actions and quantum analogue of Noether’s isomorphisms theorems. Quantum Algebra [math.QA]. Université Bourgogne Franche-Comté, 2017. English. NNT : 2017UBFCD014. tel- 01905929 HAL Id: tel-01905929 https://tel.archives-ouvertes.fr/tel-01905929 Submitted on 26 Oct 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. DOCTORAL THESIS Study of compact quantum groups with probabilistic methods: characterization of ergodic actions and quantum analogue of Noether’s isomorphisms theorems submitted by Souleiman OMAR HOCH in partial fulfillment of the requirements for the doctoral degrees in the Université de Bourgogne Franche-Comté - École Doctorale Carnot-Pasteur 29/06/2017 Advisor: Uwe FRANZ Referees: Julien BICHON and Janusz WYSOCZANSKI Examining committee: Prof. Julien BICHON Université de Clermont-Ferrand Prof. Janusz WYSOCZANSKI University of Wroclaw Prof. Eric RICARD (PRESIDENT) Université Caen-Normandie Prof. Quanhua XU Université de Bourgogne Franche-Comté Prof. Uwe FRANZ Université de Bourgogne Franche-Comté JUNE 2017 Laboratoire de Mathématiques de Besançon École doctorale Carnot-Pasteur 16 route de Gray 25030 Besançon, France Acknowledgment This thesis would never have been written without the contribution of many people. To all of them I wish to express my sincere gratitude. First of all, I would like to thank my advisor, Professor Uwe FRANZ, for his guidance and constant help during these fours years. I am delighted to have worked with him, because in addition to his scientific support, he has always been there to support me and advise me in the development of this thesis. It is a great honor that Professor Eric RICARD have accepted to chair my thesis and that Professor Quanhua XU agreed to be a member of my jury. It is a honor that Professor Julien BICHON, Professor Janusz WYSOCZAŃSKI have accepted to review my thesis. I would like to thank them all warmly. I also warmly thank all the members of the LMB (Besancon Mathematics Laboratory) in general and its functional analysis team especially among whom I have been trained rigorously during these last 4 years. I dedicate this thesis to my parents and my little family, Raysso, Leila and Soraya without whom my life would not have made sense. Your patience, endurance, sacrifices, and love keep me going. I love you all very much! Résumé Cette thèse étudie des problèmes liés aux treillis des sous-groupes quantiques et la carac- térisation des actions ergodiques et des états idempotents d’un groupe quantique compact. Elle consiste en 3 parties. La première partie présente des résultats préliminaires sur les groupes quantiques localement compacts, les sous-groupes quantiques normaux ainsi que les actions ergodiques et les états idempotents. La seconde partie étudie l’analogue quan- tique de la règle de modularité de Dedekind et de l’analogue quantique des théorèmes d’isomorphisme de Noether ainsi que leur conséquences comme le théorème de raffinement de Schreier, et le théorème Jordan-Hölder. Cette partie s’inspire du travail de recherche de Shuzhou Wang sur l’analogue quantique du troisième théorème d’isomorphisme de Noether pour les groupes quantiques compacts ainsi que le travail récent de Kasprzak, Khosravi et Soltan sur l’analogue quantique du premier théorème d’isomorphisme de Noether pour les groupes quantiques localement compacts. Dans la troisième partie, nous caractérisons les états idempotents du groupe quantique compact O−1(2) en s’appuyant sur la carac- térisation de ses actions ergodiques plongeables. Cette troisième partie est dans la ligne des travaux fait par Franz, Skalski et Tomatsu pour les groupes quantiques compacts Uq(2), SUq(2) et SOq(3). Nous classifions au préalable les actions ergodiques et les actions ergodiques plongeables du groupe quantique compact O−1(2). Les travaux présentés dans cette thèse se basent sur deux articles de l’auteur et al. Le premier s’intitule “Fundamental isomorphism theorems for quantum groups” et a été accepté pour publication dans Expositionae Mathematicae et le second est intitulé “Ergodic actions and idempotent states of O−1(2)” et est en cours de finalisation pour être soumis. Mots-clefs Groupe quantique localement compact, groupe quantique discret, groupe quantique linéaire- ment reductif, lemme de Zassenhaus, théorème de raffinement de Schreier, théorème de Jordan-Hölder, groupe quantique compact, action ergodique, état idempotent, règle de modularité de Dedekind, théorèmes d’isomorphismes de Noether. Abstract This thesis studies problems linked to the lattice of quantum subgroups and characteri- zation of ergodic actions and idempotent states of a compact quantum group. It consists of three parts. The first part present some preliminary results about locally compact quantum groups, normal quantum subgroups, ergodic actions and idempotent states. The second part studies the quantum analog of Dedekind’s modularity law, Noether’s iso- morphism theorem and their consequences as the Schreier refinement theorem and the Jordan-Hölder theorem. This part completes the work of Shuzhou WANG on the quan- tum analog of the third isomorphism theorem for compact quantum group and the recent work of Kasprzak, Khosravi and Soltan on the quantum analog of the first Noether iso- morphism theorem for locally compact quantum groups. In the third part, we characterize idempotent states of the compact quantum group O−1(2) relying on the characterization of embeddable ergodic actions. This third part is in the sequence of the seminal works of Franz, Skalski and Tomatsu for the compact quantum groups Uq(2), SUq(2) and SOq(3). We classify in advance the ergodic actions and embeddable ergodic actions of the compact quantum group O−1(2). This thesis is based on two papers of the author and al. The first one is entitled “Fundamental isomorphism theorems for quantum groups” which have been accepted for publication in Expositionae Mathematicae and the second one is entitled “Ergodic actions and idempotent states of O−1(2)” and is being finalized for submission. Keywords Locally compact quantum group, discrete quantum group, linearly reductive quantum group, Zassenhaus lemma, Schreier refinement theorem, Jordan-Hölder theorem, compact quantum group, ergodic action, idempotent state, Dedekind’s modularity law, Noether’s isomorphism theorem Contents Acknowledgment3 Introduction 11 0.1 Brève esquisse historique............................. 11 0.2 Treillis de sous-groupes quantiques....................... 15 0.3 Théorèmes fondamentaux d’isomorphismes de Noether............ 18 0.4 La règle de modularité de Dedekind et lemme de Zassenhaus........ 19 0.5 Le théorème de raffinement de Schreier et le théorème de Jordan-Hölder pour les groupes quantiques........................... 21 0.6 Caractérisation des actions ergodiques et états idempotents du groupe quan- tique compact O−1(2) .............................. 23 Introduction 29 0.1 Brief historical outline.............................. 29 0.2 Lattices of quantum subgroups......................... 33 0.3 Noether’s fundamental isomorphism theorems................. 35 0.4 Dedekind’s modular law and Zassenhaus lemma................ 37 0.5 The Schreier refinement theorem and the Jordan-Hölder theorem for quan- tum groups.................................... 38 0.6 Characterization of ergodic actions and idempotents states of the compact quantum group O−1(2) .............................. 40 1 Preliminaries 45 1.1 Preliminaries for locally compact quantum groups.............. 45 1.2 Preliminaries for linearly reductive quantum groups............. 50 1.3 Lattice of closed quantum subgroups: basic facts............... 54 1.3.1 Well positioned quantum subgroups.................. 66 1.4 Preliminaries on ergodic actions and idempotents states of a compact quan- tum group..................................... 68 2 Lattices of quantum subgroups of a linearly reductive quantum group 73 2.1 The second isomorphism theorem........................ 73 2.2 The modular law and Zassenhaus lemma.................... 76 2.3 The Schreier refinement theorem........................ 81 2.4 The Jordan-Hölder theorem........................... 82 3 Isomorphism theorems, modular law: the locally compact case 83 3.1 The second isomorphism theorem........................ 83 3.2 The third isomorphism theorem......................... 86 10 Contents 3.3 The modular law and Zassenhauss lemma................... 90 3.4 Schreier and Jordan-Hölder-type results.................... 94 4 Classification results for the compact quantum group O−1(2) 97 4.1 Ergodic actions of O−1(2) ............................ 97 4.2 Embeddable ergodic actions of O−1(2) ..................... 100 4.3 Idempotent states of O−1(2) ........................... 103 4.4 A counterexample...............................
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